INDEX 747 critical exponent, 246, 513, 553, 577, 681–688, 696 point, 117, 456, 503–508 value, 504 curvature, 213 mean, 459, 525 principal curvatures, 228 cutoff function, 330 d’Alembert’s formula, 68, 69, 78, 81, 89 Deformation Theorem, 505, 508 DeGiorgi–Moser–Nash estimates, 489 degree, 523 Derrick–Pohozaev identity, 556, 577 difference quotients, 293–295, 331, 337, 475, 486, 667, 677 and weak derivatives, 294 definition of, 294 differential equations in Banach spaces, 436 in Hilbert spaces, 506, 567 diffusion, 212, 315 and transport, 212–216 equation, 4, 44 DiPerna, R., 166 Dirac measure, 25, 35, 48, 212, 213 Dirichlet’s boundary condition, 314 energy functional, 42, 525, 526 principle, 42, 458 generalized, 458 dispersion, 177 dissipation, 177 divergence form PDE, 314, 374, 529 domain of dependence generalized wave equation, 418 nonlinear wave equation, 662–664 wave equation, 83 domain variations, 371, 513–521, 716 Dominated Convergence Theorem, 133, 153, 223, 435, 476, 548, 650, 674, 735 doubling variables, 589, 652 Douglis, A., 166 duality of Hamiltonian and Lagrangian, 121 dual space, 725 of H0 1, 301 dual variational principle, 250, 525 Duhamel’s principle, 49, 80 dynamic programming, 11, 592, 594, 607, 608 Egoroff’s Theorem, 470, 734 eigenfunction, 169, 325 basis, 357 eigenvalue, 11, 169, 325, 356, 618, 620, 725 nonsymmetric elliptic operator, 362–367 principal, 11, 358–367, 448, 549, 576 symmetric elliptic operator, 356–362 eigenvector, 356, 620, 725 left, 618 right, 618 eikonal equation, 5, 93, 606 generalized, 418 elasticity linear, 6, 89 nonlinear, 481, 502 ellipticity, 315 elliptic operator fully nonlinear, 608 linear, 314 divergence form, 314, 315, 329, 356, 367 nondivergence form, 314, 346, 362 quasilinear, 485, 543 elliptic regularization, 523 Emden–Fowler transformation, 246 energy decay wave equation, 521–522 energy estimates quasilinear elliptic equation, 532 second-order elliptic equation, 319 second-order hyperbolic equation, 404 second-order parabolic equation, 378 symmetric hyperbolic system, 428 energy flux, 663, 679, 682 energy methods heat equation, 62–65 Laplace’s equation, 41–43 wave equation, 82–84 entropy/entropy-flux pair, 648, 651, 657, 658 entropy condition, 12, 142, 148, 633, 643 condition E, 658 Lax’s, 633 Liu’s, 646 entropy solution single conservation law, 149, 165, 658 system, 649, 652

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